Topic Brief: Differentiation Rules

Provenance: written 2026-09-29. Source: research/veridian/WMA13-verified-facts.md (§1 spec table; §5 "4. Differentiation"; §6 cross-topic patterns), grepped specifically for this topic, plus this folder's own signal-frequency-and-boundaries.md (§1 frequency table), codex-deep-dive-differentiation-rules.md (§6 exam-angle synthesis) and forge-exam-technique-guide.md (§1 timing table, §5.2(d)) — read in full first so this brief adds question-number and mark-tariff synthesis those three documents don't already state, rather than repeating their citations. No new primary-source PDFs were opened for this document; every citation below traces back to one of the four files just named.

#SIGNAL Topic Master Brief: Differentiation Rules (WMA13, spec 4.1–4.3)

#Scope

This covers the same ground as the lesson differentiation-rules.ts: the five standard derivatives and their sums/differences (4.1), the product/quotient/chain rules including cosec/cot/sec (4.2), and dy/dx = 1/(dx/dy) (4.3). It excludes spec 4.4 (exponential growth/decay), which is a separate lesson and a separate frequency count.

#Every real series that has tested this topic

13 WMA13 series have been checked in the underlying research archive (8 read start-to-finish with full mark schemes and examiner reports; 5 more checked for specific topics or citation-currency only). Of those 13, this topic — 4.2 and/or 4.3 combined — has a located, citable question in 7: Oct 2020, Oct 2021, Jan 2022, Jun 2023, Jan 2024, Oct 2024, Jun 2026. Split by sub-topic: 4.2 (product/quotient/chain) is seen in 7 of 13 (the same seven series just named); 4.3 (dy/dx=1/(dx/dy)) is seen in 4 of 13, all of them a subset of that same list — Oct 2020, Jan 2022, Jun 2023, Oct 2024. No series has been found testing 4.1 in isolation — every real anchor located tests 4.1's results only as an ingredient inside a 4.2/4.3 question.

Three of the seven 4.2 citations (Oct 2021, Jan 2022 Q6a, Jan 2024 Q7a) exist in the archive only as the source of the "quote the rule" quote — the underlying question was never transcribed, so it isn't invented here.

#Where in the paper it lands

Question numbers above run Q2 to Q10 — there is no fixed slot. The strongest tendency is that this topic is rarely, if ever, the whole of a question: in every fully-detailed anchor above, differentiating is part (a) or (i), and a later part (turning point, range, vertical tangent, stationary point) uses the result. The one genuine exception to "differentiation comes mid-paper" is Oct 2024, where it opened as early as Q2 — worth knowing because numerical methods is often assumed to own the paper's early questions, and this series shows that assumption isn't safe.

#Mark-tariff pattern

• Product/quotient rule "show that f'(x) = …" alone: 4 marks, M1 A1 dM1 A1. Confirmed in this exact structure across three series — Oct 2020 Q3(a), Oct 2024 Q7(a), Jun 2026 Q3(a). The dM1 is the simplification step (combining chain-ruled terms into one fraction), not the rule application itself.
• dy/dx = 1/(dx/dy): typically ~4-5 marks, often split across two parts. Oct 2020 Q8(ii) is B1 M1 dM1 A1 (4 marks in one part); Oct 2024 Q2 splits 2+3 across two parts.
• The full question, once later parts are included, commonly reaches 8-10 marks. Oct 2024 Q7 totalled 10 marks across 4 parts once the turning-point/range work after the derivative is counted.

#Recurring context/angle

Two shapes repeat, not one:

1. A product or quotient with a root or negative-fractional-power factor, requiring the chain rule inside the product/quotient rule, then simplification to a single fraction. Confirmed twice, four years apart — Oct 2020 Q3 ($(2x+3)(4x-1)^{-1/2}$) and Oct 2024 Q7(a) ($x^3\sqrt{4x+7}$) — and independently flagged in both series' examiner reports as the discriminating step: roughly a third of candidates scored the simplification mark, a quarter scored all four.
2. dy/dx = 1/(dx/dy) used to find where a curve is "flat" in the x=f(y) sense — a turning point of x=f(y) or a vertical tangent of y=f(x). These are the same underlying event (dx/dy=0) under two different names, confirmed across Jun 2023 Q10 and Oct 2024 Q2. The Oct 2024 mark scheme records candidates instead setting the reciprocated expression equal to zero and scoring nothing for it — invalid algebra that sometimes lands on the right y-value by coincidence.

#If this topic appears, here's what to expect

• It will very likely be one sub-part of a larger question, not the whole question — expect a later part to use the derivative for something else.
• If it's phrased "show that f'(x) = [given form]," expect the mark scheme to reward the method (M1, A1 for the unsimplified form) even if the final simplification isn't reached.
• Quote the rule as its own written line before substituting. This is the single most consistently repeated piece of exam-technique advice in the whole archive for this topic — verified verbatim across 5 series (Oct 2020, Oct 2021, Jan 2022, Jun 2023, Jan 2024): "it is always advisable for candidates to quote the rule they are using before applying it to a particular function in case of slips in substitution" (Jun 2023 examiner report).
• If a root or negative-power factor is involved, expect the harder marks to be in the algebra after the rule, not the rule itself.
• If the question is dy/dx=1/(dx/dy) and later asks for a turning point or vertical tangent, work from dx/dy and set that to zero — never zero the reciprocal.
• Do not expect the derivative of arcsin, arccos or arctan — an Oct 2020 examiner report states a route using it is "beyond the specification so should not be encouraged."

#What wasn't found

No standalone spec-4.1-only question exists in the archive, and the three citation-only 4.2 anchors have no transcribed question text — both stated as gaps, not filled with invented detail. No likelihood or percentage claim is made anywhere above; every count is "seen in N of the M series checked."

#Sources

research/veridian/WMA13-verified-facts.md; signal-frequency-and-boundaries.md, codex-deep-dive-differentiation-rules.md, forge-exam-technique-guide.md (this folder).